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Vietoris–Begle Mapping Theorem
The Vietoris–Begle mapping theorem is a result in the mathematics, mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale. Theorem Let X and Y be compact space, compact metric spaces, and let f:X\to Y be surjective function, surjective and continuous function, continuous. Suppose that the image (mathematics), fibers of f are acyclic complex, acyclic, so that :\tilde H_r(f^(y)) = 0, for all 0\leq r\leq n-1 and all y\in Y, with \tilde H_r denoting the rth reduced homology, reduced Vietoris homology group (mathematics), group. Then, the induced homomorphism :f_*:\tilde H_r(X)\to\tilde H_r(Y) is an isomorphism for r\leq n-1 and a surjection for r=n. Note that as stated the theorem doesn't hold for homology theories like singular homology. For example, Vietoris homology groups of the closed topologist's sine curve and of a segment are isomorphic (since the first projects ont ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
